Find dy/dx x^3-xy+y^2=7

Problem

x3−x*y+y2=7

Solution

1. Differentiate both sides of the equation with respect to x treating y as a function of x and applying the chain rule where necessary.

d(x3)/d(x)−(d(x)*y)/d(x)+d(y2)/d(x)=d(7)/d(x)

2. Apply the power rule to the first term and the product rule to the second term.

3*x2−(xd(y)/d(x)+yd(x)/d(x))+2*yd(y)/d(x)=0

3. Simplify the expression by distributing the negative sign and noting that d(x)/d(x)=1

3*x2−xd(y)/d(x)−y+2*yd(y)/d(x)=0

4. Isolate the terms containing d(y)/d(x) on one side of the equation and move all other terms to the opposite side.

2*yd(y)/d(x)−xd(y)/d(x)=y−3*x2

5. Factor out d(y)/d(x) from the left side.

d(y)/d(x)*(2*y−x)=y−3*x2

6. Solve for d(y)/d(x) by dividing both sides by the factor (2*y−x)

d(y)/d(x)=(y−3*x2)/(2*y−x)

Final Answer

d(y)/d(x)=(y−3*x2)/(2*y−x)

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